Antennas and Their Coupling Characteristics for Wireless Power Transfer via Magnetic Coupling

ABSTRACT

Optimizing a wireless power system by separately optimizing received power and efficiency. Either one or both of received power and/ or efficiency can be optimized in a way that maintains the values to maximize transferred power.

This application claims priority from provisional application No. 61/032,061, filed Feb. 27, 2008, the disclosure of which is herewith incorporated by reference.

BACKGROUND

Our previous applications and provisional applications, including, but not limited to, U.S. patent application Ser. No. 12/018,069, filed Jan. 22, 2008, entitled “Wireless Apparatus and Methods”, the disclosure of which is herewith incorporated by reference, describe wireless transfer of power.

The transmit and receiving antennas are preferably resonant antennas, which are substantially resonant, e.g., within 10% of resonance, 15% of resonance, or 20% of resonance. The antenna is preferably of a small size to allow it to fit into a mobile, handheld device where the available space for the antenna may be limited.

An embodiment describes a high efficiency antenna for the specific characteristics and environment for the power being transmitted and received.

Antenna theory suggests that a highly efficient but small antenna will typically have a narrow band of frequencies over which it will be efficient. The special antenna described herein may be particularly useful for this kind of power transfer.

One embodiment uses an efficient power transfer between two antennas by storing energy in the near field of the transmitting antenna, rather than sending the energy into free space in the form of a travelling electromagnetic wave. This embodiment increases the quality factor (Q) of the antennas. This can reduce radiation resistance (R_(r)) and loss resistance (R_(l))

SUMMARY

The present application describes the way in which the “antennas” or coils interact with one another to couple wirelessly the power therebetween.

BRIEF DESCRIPTION OF THE DRAWINGS

In the Drawings:

FIG. 1 shows a diagram of a wireless power circuit;

FIG. 2 shows an equivalent circuit;

FIG. 3 shows a diagram of inductive coupling;

FIG. 4 shows a plot of the inductive coupling; and

FIG. 5 shows geometry of an inductive coil.

DETAILED DESCRIPTION

FIG. 1 is a block diagram of an inductively coupled energy transmission system between a source 100, and a load 150. The source includes a power supply 102 with internal impedance Z_(s) 104, a series resistance R₄ 106, a capacitance C1 108 and inductance L1 110. The LC constant of capacitor 108 and inductor 110 causes oscillation at a specified frequency.

The secondary 150 also includes an inductance L2 152 and capacitance C2 154, preferably matched to the capacitance and inductance of the primary. A series resistance R2 156. Output power is produced across terminals 160 and applied to a load ZL 165 to power that load. In this way, the power from the source 102 is coupled to the load 165 through a wireless connection shown as 120. The wireless communication is set by the mutual inductance M.

FIG. 2 shows an equivalent circuit to the transmission system of FIG. 1. The power generator 200 has internal impedance Zs 205, and a series resistance R1 210. Capacitor C1 215 and inductor L1 210 form the LC constant. A current I1 215 flows through the LC combination, which can be visualized as an equivalent source shown as 220, with a value U1.

This source induces into a corresponding equivalent power source 230 in the receiver, to create an induced power U2. The source 230 is in series with inductance L2 240, capacitance C2 242, resistance R2 244, and eventually to the load 165.

Considering these values, the equations for mutual inductance are as follows:

U₂=jωMI₁

U₁=jωMI₂

where:

z_(M)=jωM

$z_{1} = {z_{s} + R_{1} + {j\left( {{\omega \; L_{1}} - \frac{1}{\omega \; C_{1}}} \right)}}$ $z_{2} = {z_{L} + R_{2} + {j\left( {{\omega \; L_{21}} - \frac{1}{\omega \; C_{2}}} \right)}}$ z _(s) =R _(s) +jX _(s)

z _(L) =R _(L) +jX _(L)

The Mesh equations are:

$\begin{matrix} {{{U_{s} + U_{1} - {z_{1}I_{1}}} = 0}} & \rightarrow & {{I_{1} = {\left( {U_{s} + U_{1}} \right)/z_{1}}}} \\ {{{U_{2} - {z_{2}I_{2}}} = 0}} & \; & {{I_{2} = {U_{2}/z_{2}}}} \end{matrix}$ $\left. \begin{matrix} {I_{1} = \frac{U_{s} + {z_{M}I_{2}}}{z_{1}}} & {I_{2} = \frac{z_{M}I_{1}}{z_{2}}} \end{matrix}\rightarrow I_{2} \right. = {\frac{z_{M}\left( {U_{s} + {z_{M}I_{2}}} \right)}{z_{1}z_{2}} = {\left. \frac{z_{M}U_{s}}{{z_{1}z_{2}} - z_{M^{2}}}\rightarrow I_{1} \right. = {{\frac{z_{M}}{z_{M}} \cdot I_{2}} = \frac{z_{2}U_{s}}{{z_{1}z_{2}} - z_{M^{2}}}}}}$

where:

-   -   Source power:

P ₁=Re{U _(s) ·I ₁*}=U_(s)·Re{I ₁*} for avg{U _(s)}=0

-   -   Power into load:

P ₂ =I ₂ ·I ₂*Re{z _(L) }=|I ₂|²·Re{z _(L) }=|I ₂|² ·R _(L)

-   -   Transfer efficiency:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{{I_{2} \cdot I_{2}^{\star}}R_{L}}{U_{s}{Re}\left\{ I_{1}^{\star} \right\}}}$ ${I_{2} \cdot I_{2}^{\star}} = \frac{z_{M}z_{M}^{\star}U_{s}^{2}}{\left( {{z_{1}z_{2}} - z_{M^{2}}} \right)\left( {{z_{1}^{\star}z_{2}^{\star}} - z_{M^{2}}^{\star}} \right)}$ ${{Re}\left\{ I_{1}^{\star} \right\}} = {{Re}\left\{ \frac{z_{2}^{\star}U_{s}}{{z_{1}^{\star}z^{\star}} - z_{M^{2}}^{\star}} \right\}}$

Overall transfer Efficiency is therefore:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{{U_{s}^{2} \cdot R_{L}}z_{M}z_{M}^{\star}}{\left( {{z_{1}z_{2}} - z_{M}^{2}} \right)\left( {{z_{1}^{\star}z_{2}^{\star}} - z_{M^{2}}^{\star}} \right)}}$

Def.: z ¹ =z ₁ z ₂ −z _(M) ^(si 2)

$\left. \begin{matrix} {\left. \rightarrow\eta \right. = {\frac{P_{2}}{P_{1}} = {\frac{R_{L}z_{M}z_{M}^{\star}}{z^{\prime}z^{\star}{Re}\left\{ \frac{z_{2}^{\star}z^{\prime}}{z^{\prime}z^{\prime \star}} \right\}} = \frac{R_{L}z_{M}z_{M}^{\star}}{{Re}\left\{ {z_{2}^{\star} \cdot z^{\prime}} \right\}}}}} \\ {= {\frac{R_{L}z_{M}z_{M}^{\star}}{{Re}\left\{ {z_{2}^{\star}\left( {{z_{1}z_{2}} - z_{M}^{2}} \right)} \right\}} = \frac{R_{L}{z_{M}}^{2}}{{Re}\left\{ {{z_{1}{z_{2}}^{2}} - {z_{2}^{\star}z_{M}^{2}}} \right\}}}} \end{matrix}\rightarrow\eta \right. = {\frac{P_{2}}{P_{1}} = \frac{R_{L}{z_{M}}^{2}}{{{{z_{2}}^{2} \cdot {Re}}\left\{ z_{1} \right\}} - {z_{M}^{2}{Re}\left\{ z_{2}^{\star} \right\}}}}$ Re{z₁} = R_(s) + R₁ Re{z₂^(⋆)} = R_(L) + R₂ ${z_{2}}^{2} = {\left( {R_{L} + R_{2}} \right)^{2} + \left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + X_{L}} \right)^{2}}$ z_(M)² = ω²M² z_(M²) = (j ω M)² = −ω²M²

A Transfer efficiency equation can therefore be expressed as:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{\omega^{2}{M^{2} \cdot R_{L}}}{{\left( {R_{s} + R_{n}} \right)\begin{bmatrix} {\left( {R_{L} + R_{2}} \right)^{2} +} \\ \left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + X_{L}} \right) \end{bmatrix}} + {\omega^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}}$

Which reduces in special cases as follows: A) when ω=ω₀=1/√{square root over (L₂C₂)}, X_(L)=0 or where

${{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + {X_{L}(\omega)}} = 0$ $\begin{matrix} {\eta = \frac{P_{2}}{P_{1}}} \\ {= {\frac{\omega_{0}^{2}M^{2}}{\left\lbrack {{\left( {R_{s} + R_{n}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M_{2}}} \right\rbrack} \cdot \frac{R_{L}}{\left( {R_{L} + R_{2}} \right)}}} \end{matrix}$

B) when ω=ω₀, R_(s)=0:

$\begin{matrix} {\eta = \frac{P_{2}}{P_{1}}} \\ {= \frac{\omega_{0}^{2}M^{2}R_{L}}{{R_{1}\left( {R_{L} + R_{2}} \right)}^{2} + {\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}} \end{matrix}$

C) when ω=ω₀, R_(s)=0 R_(L)=R₂:

$\begin{matrix} {\eta = \frac{P_{2}}{P_{1}}} \\ {= \frac{\omega_{0}^{2}M^{2}}{{4R_{1}R_{2}} + {2\omega_{0}^{2}M^{2}}}} \end{matrix}$

D) when ω=ω₀, R_(s)=0 R_(L)=R₂ 2R₁R₂>>ω₀ ²M²:

$\eta = {\frac{P_{2}}{P_{1}} \cong {\frac{\omega_{0}^{2}M^{2}}{4R_{1}R_{2}}\left( {{weak}\mspace{14mu} {coupling}} \right)}}$

where: Mutual inductance:

M=k√{square root over (L ₁ L ₂)} where k is the coupling factor

Loaded Q factors:

$Q_{1,L} = \frac{\omega \; L_{1}}{R_{s} + R_{1}}$ $Q_{2,L} = \frac{\omega \; L_{2}}{R_{L} + R_{2}}$

Therefore, the transfer efficiency in terms of these new definitions: A) when ω=ω₀

$\begin{matrix} {\eta = \frac{P_{2}}{P_{1}}} \\ {= {\frac{k^{2} \cdot \frac{\omega_{0}{L \cdot \omega_{0}}L_{2}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)}}{1 + {k^{2} \cdot \frac{\omega_{0}{L \cdot \omega_{0}}L_{2}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)}}} \cdot \frac{R_{L}}{R_{L} + R_{2}}}} \\ {\eta = {\frac{k^{2} \cdot Q_{1,L} \cdot Q_{2,L}}{1 + {k^{2} \cdot Q_{1,L} \cdot Q_{2,L}}} \cdot \frac{R_{L}}{R_{L} + R_{2}}}} \end{matrix}$

C) when ω=ω₀, R_(L)=R₂,(R_(s)=0):

D) ω=ω₀, R_(L)=R₂, (R_(s)=0), 2R_(n)R₂>>ω₀ ²M²

→1>>k²Q_(1,UL)Q_(2,UL)/2

$\eta = {\frac{P_{2}}{P_{n}} \cong {\frac{k^{2}Q_{1,{UL}}Q_{2,{UL}}}{4}\left( {{weak}\mspace{14mu} {coupling}} \right)}}$

Q_(UL): Q unloaded

${Q_{1,{UL}} = \frac{\omega \; L_{1}}{R_{1}}};$ $Q_{2,{UL}} = \frac{\omega \; L_{2}}{R_{2}}$

This shows that the output power is a function of input voltage squared

P₂ = f(U_(s)²) + I₂ ⋅ I₂^(*)R_(L); $I_{2} = \frac{z_{M}U_{s}}{{z_{1}z_{2}} - z_{M^{2}}}$ $P_{2} = {\frac{z_{M}z_{M}^{*}R_{L}}{\left( {{z_{1}z_{2}} - z_{M}^{2}} \right)\left( {{z_{1}^{*}z_{2}^{*}} - z_{M}^{2^{*}}} \right)} \cdot U_{s}^{2}}$ $P_{2} = \frac{{z_{M}}^{2} \cdot R_{L} \cdot U_{s}^{2}}{{z_{1}z_{2}z_{1}^{*}z_{2}^{*}} + {z_{M}} + {{z_{M}}^{2} \cdot \left( {{z_{1}z_{2}} + {z_{1}^{\;^{*}}z_{2}^{*}}} \right)}}$ $P_{2} = \frac{{z_{M}}^{2} \cdot R_{L} \cdot U_{s}^{2}}{{{z_{1}z_{2}}}^{2} + {{z_{M}}^{2}2{Re}\left\{ {z_{1}z_{2}} \right\}} + {z_{M}}^{4}}$ z_(M) = jω M z_(M)^(*) = −jω M z_(M) = ω M = z_(M)z_(M)^(*) z_(M)^(*) = −ω ²M² = −z_(M)² z_(M)^(2^(*)) = −ω ²M² = z_(M)² = −z_(M)² z_(M)² ⋅ z_(M)^(2^(*)) = z_(M)⁴ z₁z₂ = z₁ ⋅ z₂ z₁z₂ + z₁^(*)z₂^(*) = 2Re{z₁z₂} z₁ ⋅ z₂² = z₁² ⋅ z₂²

Definitions:

z ₁ R′ ₁ +jX ₁ ; z ₂ =R′ ₂ +jX ₂

|z ₁ z ₂|²=(R′ ₁ ² +X ₁ ²)(R′ ₂ ² +X ₂ ²)=R′ ₁ ² R′ ₂ ² +X ₁ ² R′ ₂ ² +X ₂ ² R′ ₁ ² +X ₁ ² X ₂ ²

Re{z ₁ z ₂}=Re(R′ ₁ +jX ₁)(R′ ₂ +jX ₂)=R′ ₁ R′ ₂ +X ₁ X ₂

|z _(M) |=X _(M)

$P_{2} = \frac{X_{M}^{2}{R_{1} \cdot U_{s}^{2}}}{\begin{matrix} {{R_{1}^{\prime 2}R_{2}^{\prime 2}} + {R_{1}^{\prime 2}X_{2}^{2}} + {R_{1}^{\prime 2}X_{1}^{2}} + {X_{1}^{2}X_{2}^{2}} +} \\ {{2\; X_{M}^{2}R_{1}^{\prime}R_{2}^{\prime}} + {2\; X_{M}^{2}X_{1}X_{2}} + X_{M}^{4}} \end{matrix}}$ $P_{2} = \frac{X_{M}^{2}{R_{L} \cdot U_{s}^{2}}}{\left( {{R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{2}} \right)^{2} + {R_{1}^{\prime 2}X_{2}^{2}} + {R_{2}^{\prime 2}X_{1}^{2}} + {X_{1}^{2}X_{2}^{2}} + {2X_{M}^{2}X_{1}X_{2}}}$

Therefore, when at or near the resonance condition:

ω=ω₀=ω₂=ω₀→X₁=0, X₂=0

$\begin{matrix} {P_{2} = \frac{X_{M}^{2}{R_{1} \cdot U_{s}^{2}}}{{R_{1}^{\prime 2}R_{2}^{\prime 2}} + {2\; X_{M}^{2}R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{4}}} \\ {= {\frac{X_{M}^{2}R_{L}}{\left( {{R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{2}} \right)^{2}} \cdot U_{s}^{2}}} \\ {P_{2} = {\frac{\omega_{0}^{2}M^{2}R_{L}}{\begin{matrix} {{\left( {R_{s} + R_{1}} \right)^{2}\left( {R_{1} + R_{2}} \right)^{2}} + {2\omega_{0}^{2}M^{2}}} \\ {{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{4}M^{4}}} \end{matrix}} \cdot U_{s}^{2}}} \\ {P_{2} = {\frac{\omega_{0}^{2}M^{2}R_{L}}{\left( {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right)^{2}}U_{s}^{2}}} \end{matrix}$

Showing that the power transfer is inversely proportional to several variables, including series resistances.

Mutual inductance in terms of coupling factors and inductions:

$M = {k \cdot \sqrt{L_{1}L_{2}}}$ $\begin{matrix} {P_{2} = {\frac{\omega_{0}^{2}k^{2}L_{1}{L_{2} \cdot R_{L}}}{\left( {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}k^{2}L_{1}L_{2}}} \right)^{2}} \cdot U_{s}^{2}}} \\ {= {\frac{k^{2}\frac{\omega_{0}L_{1}\omega_{0}L_{2}}{\left( {R_{s} + R_{M}} \right)\left( {R_{1} + R_{2}} \right)}}{\left( {1 + {k^{2}\frac{\omega_{0}L_{1}\omega_{0}L_{2}}{\left( {R_{s} + R_{M}} \right)\left( {R_{1} + R_{2}} \right)}}} \right)^{2}} \cdot \frac{U_{s}^{2}R_{L}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)}}} \\ {P_{2} = {\frac{k^{2} \cdot Q_{L\; 1} \cdot Q_{L\; 2}}{\left( {1 + {k^{2} \cdot Q_{L\; 1} \cdot Q_{L\; 2}}} \right)^{2}} \cdot \frac{R_{L}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} \cdot U_{s}^{2}}} \end{matrix}$

The power output is proportional to the square of the input power, as described above. However, there is a maximum input power beyond which no further output power will be produced. These values are explained below. The maximum input power P1max is expressed as:

${P_{1,\max} = {\frac{U_{s}^{2}}{R_{s} + R_{{in},\min}} = {{Re}\left\{ {U_{s} \cdot I_{1}^{*}} \right\}}}};$

R_(in,min): permissible input resistance Efficiency relative to maximum input power:

$\begin{matrix} {\eta^{\prime} = \frac{P_{2}}{P_{1,\max}}} \\ {= \frac{P_{2}\left( U_{s}^{2} \right)}{P_{1,\max}}} \end{matrix}$

Under resonance condition ω=ω₁=ω₂=ω₀:

$\eta^{\prime} = \frac{\omega_{0}^{2}M^{2}{R_{L}\left( {R_{s} + R_{{in},\min}} \right)}}{\left\lbrack {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack^{2}}$

Equation for input power (P₁) under the resonance condition is therefore.

$P_{1} = {\frac{P_{2}}{\eta} = {\frac{\omega_{0}^{2}M^{2}{R_{L}\left\lbrack {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack}\left( {R_{L} + R_{2}} \right)}{{\left\lbrack {{\left( {R_{s} + R_{2}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{0}M^{2}}} \right\rbrack^{2} \cdot \omega_{0}^{2}}M^{2}R_{L}} \cdot U_{s}^{2}}}$ $P_{1} = {\frac{R_{L} + R_{2}}{{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} +_{0}^{2}M^{2}} \cdot U_{S}^{2}}$ ${{For}\mspace{14mu} \left( {R_{s} + R_{M}} \right)\left( {R_{L} + R_{2}} \right)}\operatorname{>>}{\omega_{0}^{2}{M^{2}:{P_{1} \cong \frac{U_{S}^{2}}{\left( {R_{s} + R_{1}} \right)}}}}$

The current ratio between input and induced currents can be expressed as

$\frac{I_{2}}{I_{1}} = {\frac{z_{M} \cdot U_{s} \cdot \left( {{z_{1}z_{2}} - z_{n}^{2}} \right)}{\left( {{z_{1}z_{2}} - z_{M}^{2}} \right)z_{2}U_{s}} = {\frac{z_{M}}{z_{2}} = \frac{j\; \omega \; M}{R_{L} + R_{2} + {j\left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}}} \right)}}}}$ ${{at}\mspace{14mu} \omega} = {\omega_{0} = \frac{1}{\sqrt{L_{2}C_{2}}}}$ $\frac{I_{2}}{I_{1}} = {{\frac{j\; \omega \; M}{R_{1} + R_{2}}\mspace{14mu} {{avg}.\left\{ \frac{I_{2}}{I_{1}} \right\}}} = \frac{\pi}{2}}$

Weak coupling: R1+R₂>|jωM|

→I₂<I₁

Strong coupling: R₁+R₂<|jωM|

→I₂>I₁

Input current I: (under resonance condition)

$I_{1} = {\frac{P_{1}}{U_{S}} = \frac{\left( {R_{1} + R_{2}} \right) \cdot U_{s}}{{\left( {R_{S} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}}}$ $I_{1} = {\frac{\left( {R_{L} + R_{2}} \right)}{{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \cdot U_{s}}$

Output current I₂: (under resonance condition)

$I_{2} = {\frac{j\; \omega \; M}{{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \cdot U_{s}}$

Maximizing transfer efficiency and output power (P₂)

can be calculated according to the transfer efficiency equation:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{\omega^{2}M^{2}R_{L}}{{\left( {R_{s} + R_{n}} \right)\begin{bmatrix} {\left( {R_{L} + R_{2}} \right)^{2} +} \\ \underset{}{\left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + X_{L}} \right)^{2}} \end{bmatrix}} + {\omega^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}}$

After reviewing this equation, an embodiment forms circuits that are based on observations about the nature of how to maximize efficiency in such a system.

Conclusion 1)

η(L₂, C₂, X_(L)) reaches maximum for

${{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + X_{L}} = 0$

That is, efficiency for any L, C, X at the receiver is maximum when that equation is met. Transfer efficiency wide resonance condition:

$\eta = {{\frac{P_{2}}{P_{1}}_{\omega = \omega_{0}}} = {\frac{\omega_{0}^{2}M^{2}}{\left\lbrack {{\left( {\underset{}{R_{s}} + R_{n}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack} \cdot \frac{R_{1}}{\left( {R_{L} + R_{2}} \right)}}}$

Conclusion 2)

To maximise η R_(s) should be R_(s)<<R₁ That is, for maximum efficiency, the source resistance Rs needs to be much lower than the series resistance, e.g., 1/50, or 1/100^(th) or less Transfer efficiency under resonance and weak coupling condition:

(R_(s) + R_(n))(R_(L) + R₂)>> ω₀²M² $\eta \cong \frac{\omega_{0}^{2}{M^{2} \cdot \overset{}{R_{L}}}}{\left( {R_{s} + R_{n}} \right)\left( {\underset{}{R_{L}} + R_{2}} \right)^{2}}$

Maximising η(R_(L)):

$\frac{\eta}{R_{L}} = {{\frac{\omega_{0}^{2}M^{2}}{R_{s} + R_{1}} \cdot \frac{\left( {R_{L} + R_{2}} \right) - {2\; R_{L}}}{\left( {R_{L} + R_{2}} \right)^{3}}} = {\left. 0\rightarrow R_{L} \right. = R_{2}}}$

Conclusion 3)

η reaches maximum for R_(L)=R₂ under weak coupling condition. That is, when there is weak coupling, efficiency is maximum when the resistance of the load matches the series resistance of the receiver. Transfer efficiency under resonance condition. Optimising R_(L) to achieve max. η

${\frac{\eta}{R_{L}} = 0};{{\frac{}{R_{L}} \cdot \frac{\omega_{0}^{2}M^{2}R_{L}}{{\underset{\underset{R_{1}}{}}{\left( {R_{s} + R_{1}} \right)}\left( {R_{L} + R_{2}} \right)^{2}} + {\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}}\frac{u}{v}}$ $\frac{{u \cdot v^{\prime}} - {v \cdot u^{\prime}}}{v^{2}} = 0$ u = ω₀²M² ⋅ R_(L); u^(′) = ω₀²M² v = R₁^(′)(R_(L) + R₂)² + ω₀²M²(R₁ + R₂) u ⋅ v^(′) − v ⋅ u^(′) = 0 v^(′) − 2 R₁^(′)(R_(L) + R₂) + ω₀²M² $\begin{matrix} {{{u \cdot v^{\prime}} - {v \cdot u^{\prime}}} = {{\text{?}\left( {{2{R_{1}^{\prime}\left( {R_{L} + R_{2}} \right)}} + {\omega_{0}^{2}M^{2}}} \right)} -}} \\ {{\left( {{R_{1}^{\prime}\left( {R_{1} + R_{2}} \right)}^{2} + {\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}} \right)\text{?}}} \\ {= 0} \\ {= {{2\; R_{1}^{\prime}{R_{L}\left( {R_{L} + R_{2}} \right)}} + {\omega_{0}^{2}M^{2}R_{L}} -}} \\ {{{R_{1}^{\prime}\left( {R_{L} + R_{2}} \right)}^{2} - {\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}} \\ {= 0} \\ {= {\text{?} + \text{?} + \text{?} - \text{?} - \text{?} - \text{?} - \text{?} - \text{?}}} \\ {= 0} \\ {= {{\left( {{1\; R_{1}^{\prime}} - R_{1}^{\prime}} \right)R_{L}^{2}} - {R_{1}^{\prime}R_{2}^{2}} - {\omega_{0}^{2}M^{2}R_{2}}}} \\ {= 0} \end{matrix}$ $R_{L}^{2} = \frac{{R_{1}^{\prime}R_{2}^{2}} + {\omega_{0}^{2}M^{2}R_{2}}}{R_{1}^{\prime}}$ $\begin{matrix} {R_{L} = {\pm \sqrt{\frac{{\left( {R_{s} + R_{1}} \right)R_{2}^{2}} + {\omega_{0}^{2}M^{2}R^{2}}}{\left( {R_{s} + R_{1}} \right)}}}} \\ {= {{\pm R_{2}} \cdot \sqrt{\frac{\left( {R_{s} + R_{1}} \right) + {\omega_{0}^{2}{M^{2}/R^{2}}}}{\left( {R_{s} + R_{1}} \right)}}}} \end{matrix}$ $\begin{matrix} {R_{L,{opt}} = {R_{2}\sqrt{1 + \frac{\omega_{0}^{2}M^{2}}{\left( {R_{s} + R_{1}} \right)R_{2}}}}} & \; \end{matrix}$ ?indicates text missing or illegible when filed

Weak coupling condition ω₀ ²M²<<(R_(s)+R₁)R₂

Conclusion 4)

There exists an optimum R_(L)>R₂ maximising η Output power P₂:

$P_{2} = \frac{X_{M}^{2}R_{1}U_{s}^{w}}{\left( {{R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{2}} \right)^{2} + {R_{1}^{\prime 2}\underset{}{X_{2}^{2}}} + {R_{2}^{\prime 2}\underset{}{X_{1}^{2}}} + \underset{}{X_{1}^{2}X_{2}^{2}} + {2X_{M}^{2}\underset{}{X_{2}X_{2}}}}$

Conclusion 5)

Output power P₂(X₁, X₂) reaches maximum for

$X_{1} = {{{\omega \; L_{1}} - \frac{1}{\omega \; C_{1}} + X_{s}} = 0}$ $X_{2} = {{{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + X_{L}} = 0}$

that is, when there is a resonance condition at both the primary and the secondary. Output power P₂ wide resonance condition:

$P_{2} = {\frac{\omega_{0}^{2}{M^{2} \cdot R_{L}}}{\left\lbrack {{\left( {\underset{}{P_{s}} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack^{2}} \cdot U_{s}^{2}}$

Conclusion 6)

To maximize P₂, R_(s) should be R_(s)<<R₁ Output power P₂ for the wide resonance and weak coupling condition:

(R _(s) +R ₁)(R _(L) +R ₂)>>ω₀ ² M ²

$P_{2} \cong {\frac{\omega_{0}^{2}{M^{2} \cdot R_{L}}}{\left( {R_{s} + R_{1}} \right)^{2} + \left( {R_{L} + R_{2}} \right)^{1}} \cdot U_{s}^{2}}$

Conclusion 7)

P₂ (R_(L)) reaches maximum for R_(L)=R₂ (see conclusion 3) For each of the above, the >> or << can represent much greater, much less, e.g., 20× or 1/20 or less, or 50× or 1/50^(th) or less or 100× or 1/100^(th) or less. The value R_(L) can also be optimized to maximize P₂:

$\frac{P_{2}}{R_{L}} = 0$ $\frac{{u \cdot v^{\prime}} - {v \cdot u^{\prime}}}{v^{2}} = 0$ u = ω₀²M²R_(L); u^(′) = ω₀²M² v = [(R₁^(′))(R_(L) + R₂) + ω₀²M²]² v^(′) = 2 ⋅ [R₁^(′)(R_(L) + R₂) + ω₀²M²] ⋅ R₁^(′) ${{{\text{?} \cdot R_{L} \cdot {2\begin{bmatrix} {{R_{1}^{\prime}\left( {R_{1} + R_{2}} \right)} +} \\ {\omega_{0}^{2}M^{2}} \end{bmatrix}}}R_{1}} - {\begin{bmatrix} {{R_{1}^{\prime}\left( {R_{L} + R_{2}} \right)} +} \\ {\omega_{0}^{2}M^{2}} \end{bmatrix}^{2} \cdot \text{?}}} = 0$ 2 R_(L)(R₁^(′2)R_(L) + R₁^(′2)R₂) + 1 R_(L)ω₀²M² ⋅ R₁^(′) − [R₁^(′)R_(L) + R₁^(′)R₂ + ω₀²M²]² = 0 2? + ? + ? − ? − R₁^(′2)R₂² − ω₀²M⁴ − ? − 2? − 2R₁^(′)R₂ω₀²M² = 0 = (2R₁^(′2) − R₁^(′2))R_(L)² − R₁^(′2)R₂² − 2 R₁^(′)R₂ω₀²M² − ω₀²M⁴ = 0 = R₁^(′2) ⋅ R_(L)² − (R₁^(′)R₂ + ω₀²M²)² = 0 $R_{L}^{2} = \frac{\left( {{R_{1}^{\prime}R_{2}} + {\omega_{0}^{2}M^{2}}} \right)^{2}}{R_{1}^{\prime 2}}$ $\begin{matrix} {R_{L,{opt}} = \frac{{R_{1}^{\prime}R_{2}} + {\omega_{0}^{2}M^{2}}}{R_{1}^{\prime}}} \\ {= {R_{2}\left( {1 + \frac{\omega_{0}^{2}M^{2}}{\left( {R_{s} + R_{1}} \right)R_{2}}} \right)}} \end{matrix}$ $\begin{matrix} {R_{L,{opt}} = {R_{2} \cdot \left( {1 + \frac{\omega_{0}^{2}M^{2}}{\left( {R_{s} + R_{1}} \right)R_{2}}} \right)}} & \; \end{matrix}$ $\begin{matrix} {{Weak}\mspace{14mu} {coupling}\text{:}R_{L,{opt}}\begin{matrix}  \cong \\  >  \end{matrix}R_{2}} & \; \end{matrix}$ ?indicates text missing or illegible when filed

Conclusion 8)

There exists an optimum R_(L)>R₂ maximising P₂. This R_(1opt) differs from the R_(1,opt) maximising η. One embodiment operates by optimizing one or more of these values, to form an optimum value.

Inductive coupling is shown with reference to FIGS. 3, 4 FIG. 5 illustrates the Inductance of a multi-turn circular loop coil

$R_{m} = \frac{R_{0} + R_{1}}{2}$

Wheeler formula (empirical) $L = \frac{0.8\; {R_{m}^{2} \cdot N^{2}}}{{6\; R_{m}} + {9\; w} + {10\left( \left( {R_{0} - R_{1}} \right) \right.}}$ [Wheeler, H. A., “Simple inductance formulas for radio coils”. Proc. IRE Vol 16, pp. 1328-1400, October 1928.] Note: this i accurate if all three terms in denominator are about equal. Conversion to H, m units: [L]μH $L = \frac{0.8 \cdot R_{m}^{2} \cdot \wp^{2} \cdot N^{1} \cdot 10^{- 6}}{{6\; {R_{m} \cdot \wp}} + {9 \cdot w \cdot \wp} + {10\left( {R_{0} - R_{1}} \right)\wp}}$ $L = \frac{0.8 \cdot R_{m}^{2} \cdot \wp^{2} \cdot N^{2} \cdot 10^{- 6}}{{6\; R_{m}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}}$ [R_(m), R_(i), R₀, ω] = inch ${1\; m} = {\frac{\overset{\wp}{\overset{}{10}}00}{- 154} \cdot {inch}}$ 1H = 10⁶ μH [L] = H [R_(m)R₀R₁ω] = m

In standard form:

${L = \frac{\mu_{0} \cdot A_{m} \cdot N^{2}}{K_{c}}};$ A_(m) = π ⋅ R_(m)² μ₀ = 4 π ⋅ 10⁻⁷ $L = \frac{0.8 \cdot \cdot 10^{- 6} \cdot {\overset{}{\pi \; R}}_{m}^{2} \cdot N^{2} \cdot \overset{}{4\; {\pi \cdot 10^{- 7}}}}{{\pi \cdot 4}\; {\pi \cdot 10^{- 7}}\left( {{6\; R_{m}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}$ $L = {\frac{\mu_{0}{A_{m} \cdot N^{2} \cdot 0.8}{\text{?} \cdot 10}}{4\; {\pi^{2} \cdot \text{?}}\left( {{6\; R_{m}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}\frac{1}{K_{c}}}$ $R_{m} = \sqrt{\frac{A_{m}}{\pi}}$ $K_{c} = \frac{{\pi^{2} \cdot 25.4}\left( {{6\sqrt{\frac{A_{m}}{\pi}}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}{2 \cdot 1000}$ $K_{c} \cong {\frac{1}{8} \cdot \left( {{6\sqrt{\frac{A_{m}}{\pi}}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}$ ${L = \frac{\mu_{0}A_{m}N^{2}}{K_{c}}};$ $A_{m} = {{\left( \frac{\left( {R_{0} + R_{1}} \right)}{2} \right)^{2} \cdot {\pi \lbrack L\rbrack}} = H}$ ?indicates text missing or illegible when filed

The inductance of a single-turn circular loop is given as:

$K_{c} = \frac{R_{m} \cdot \pi}{\left\lbrack {\frac{8\; R_{m}}{6} - 2} \right\rbrack}$ ${L = \frac{\mu_{0}A_{m}}{K_{c}}};{A_{m} = {{R_{m}^{2} \cdot {\pi \lbrack L\rbrack}} = H}}$

where:

R_(m): mean radius in m

b: wire radius in m,

For a Numerical example:

R₁=0.13 m

R₀=0.14 m

ω=0.01 m

N=36

→L=0.746 mH

The measured inductance

L_(meas)=0.085 mH

The model fraction of Wheeler formula for inductors of similar geometry, e.g, with similar radius and width ratios is:

$K_{c} = {\frac{1}{8}\left( {{5\sqrt{\frac{A_{m}}{\pi}}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}$ $D = \sqrt{W^{2} + \left( {R_{0} - R_{1}} \right)^{2}}$ $R_{m} = \frac{R_{0} + R_{1}}{2}$

Using a known formula from Goddam, V. R., which is valid for

w>(R ₀ −R ₁)

$L = {{0.03193 \cdot R_{m} \cdot N^{2}}\left\lfloor \begin{matrix} {{2.303\left( {1 + \frac{w^{2}}{32\; R_{m}^{2}} + \frac{D^{2}}{96\; R_{m}^{2}}} \right){\log \left( \frac{8\; R_{m}}{D} \right)}} -} \\

\end{matrix} \right\rfloor}$

1w H,m units:

$L = {\mu_{0}{R_{m} \cdot N^{2}}\left\lfloor \begin{matrix} {{\left( {1 + \frac{w^{2}}{32\; R_{m}^{2}} + \frac{D^{2}}{96\; R_{m}^{2}}} \right){n\left( \frac{8\; R_{m}}{D} \right)}} -} \\

\end{matrix} \right\rfloor}$

Example 1:

R₁ = 0.13 m R₀ = 0.14 m W = 0.01 m N = 36 L = 757 μH ${{Ratio}\text{:}\mspace{11mu} \frac{W}{R_{0} - R_{1}}} = 1$ → y₁ = 0.8483    y₂ = 0.816 From [Terman, F.]

Example 2: (given in [Goddam, V. R.]

R₀ = 8.175 inches R₁ = 7.875 inches W = 2 inches N = 57 y₁ = 0.6310 y₂ = 0.142 → L = 2.5 mH (2.36 mH) ${Ratio}\text{:}\mspace{11mu} \begin{matrix} {\frac{2}{R_{0} - R_{1}} = {\frac{2}{0.3} = {6.667\mspace{14mu} {or}}}} \\ {{\frac{R_{0} - R_{1}}{W} = {\frac{0.3}{2} = 0.15}}\mspace{45mu}} \end{matrix}$

where Goddam, V. R. is the Thesis Masters Louisiana State University, 2005, and Terman, F. is the Radio Engineers Handbook, McGraw Hill, 1943.

Any of these values can be used to optimize wireless power transfer between a source and receiver.

From the above, it can be seen that there are really two different features to consider and optimize in wireless transfer circuits. A first feature relates to the way in which efficiency of power transfer is optimized. A second feature relates to maximizing the received amount of power—independent of the efficiency.

One embodiment, determines both maximum efficiency, and maximum received power, and determines which one to use, and/or how to balance between the two.

In one embodiment, rules are set. For example, the rules may specify:

Rule 1—Maximize efficiency, unless power transfer will be less than 1 watt. If so, increase power transfer at cost of less efficiency.

Rule 2—Maximize power transfer, unless efficiency becomes less than 30%.

Any of these rules may be used as design rules, or as rules to vary parameters of the circuit during its operation. In one embodiment, the circuit values are adaptively changes based on operational parameters. This may use variable components, such as variable resistors, capacitors, inductors, and/or FPGAs for variation in circuit values.

Although only a few embodiments have been disclosed in detail above, other embodiments are possible and the inventors intend these to be encompassed within this specification. The specification describes specific examples to accomplish a more general goal that may be accomplished in another way. This disclosure is intended to be exemplary, and the claims are intended to cover any modification or alternative which might be predictable to a person having ordinary skill in the art. For example, other sizes, materials and connections can be used. Other structures can be used to receive the magnetic field. In general, an electric field can be used in place of the magnetic field, as the primary coupling mechanism. Other kinds of antennas can be used. Also, the inventors intend that only those claims which use the-words “means for” are intended to be interpreted under 35 USC 112, sixth paragraph. Moreover, no limitations from the specification are intended to be read into any claims, unless those limitations are expressly included in the claims.

Where a specific numerical value is mentioned herein, it should be considered that the value may be increased or decreased by 20%, while still staying within the teachings of the present application, unless some different range is specifically mentioned. Where a specified logical sense is used, the opposite logical sense is also intended to be encompassed. 

1. A method of forming a wireless power system, comprising: first optimizing efficiency of power transfer between a transmitter of power to a receiver of wireless power; and separate from said optimizing efficiency, second optimizing maximum received power in said receiver.
 2. A method as in claim 1, wherein said first optimizing and said second optimizing are done according to rules which specify information about both desired efficiency and desired total received power.
 3. A method as in claim 2, wherein said rules specify a minimum efficiency for power transfer.
 4. A method as in claim 2, wherein said rules specify a minimum received power amount.
 5. A method as in claim 1, wherein said optimizing comprises optimizing multiple different aspects simultaneously, including a first aspect that relates to comparing resonance in a source with a resonance in the receiver, and a second aspect that relates to a strength of a coupling between said transmitter and said receiver.
 6. A method as in claim 5, wherein said optimizing is carried out separately for a weaker coupling as compared with a stronger coupling.
 7. A method as in claim 5, wherein said optimizing comprises maintaining resonance conditions of both said transmitter and said receiver.
 8. A method as in claim 5, wherein said optimizing further comprises maintaining a resistance of an inductor in the receiver substantially equal to a series resistance.
 9. A method as in claim 5, wherein said optimizing further comprises maintaining a source resistance at a transmitter as less than a series resistance of the transmitter.
 10. A system, comprising: a receiver of a wireless power, including and inductor, a capacitor, and a connection to a load, wherein said receiver has values that are optimized, according to both of power transmitter between a remote transmitter of power that wirelessly transmits the power to the receiver, and also according to a maximum receive power efficiency, that optimizes the maximum received power in the receiver.
 11. A system as in claim 10, wherein said receiver has first optimizing and said second optimizing which are done according to rules which specify information about both desired efficiency and desired total received power.
 12. A method as in claim 11, wherein said rules specify a minimum efficiency for power transfer.
 13. A method as in claim 11, wherein said rules specify a minimum received power amount.
 14. A method as in claim 11, wherein said rules optimize multiple different aspects simultaneously, including a first aspect that relates to comparing resonance in a source with a resonance in the receiver, and a second aspect that relates to a strength of a coupling between said transmitter and said receiver.
 15. A method as in claim 14, wherein said optimizing is carried out separately for a weaker coupling as compared with a stronger coupling.
 16. A method as in claim 14, wherein said optimizing comprises maintaining resonance conditions of both said transmitter and said receiver.
 17. A method as in claim 14, wherein said optimizing further comprises maintaining a resistance of an inductor in the receiver substantially equal to a series resistance.
 18. A method as in claim 14, wherein said optimizing further comprises maintaining a source resistance at a transmitter as less than a series resistance of the transmitter. 